Abstract
In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant.
This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.
Citation
Kenji Fukaya. "Counting pseudo-holomorphic discs in Calabi-Yau 3-holds." Tohoku Math. J. (2) 63 (4) 697 - 727, 2011. https://doi.org/10.2748/tmj/1325886287
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